We used the provided shape and velocity parameters to integrate back in time the position and movement of each of the four spiral arms. In parallel, we integrated the orbital motion of each HMXB around the Milky Way using their currently observed astrometric parameters from Gaia EDR3. The integration is performed with the python package Galpy along with the galactic potential solution MWPotential2014 from [1]} with no additional wrapper. This 2D potential takes into account the contribution of the Galactic bulge, disk and a surrounding dark matter halo; neither the central black hole nor the central bar are part of the model, but we did not investigate any source that comes anywhere close to the Galactic center. Another caveat could be the presence of a Galactic warp, which precession was recently quantified in [2]} and may cause a perturbation in the potential that we did not take into account here. The set of parameters we used for the normalisation of the Galactic rotation curve are R\(_{\odot }\) =8.178 kpc [3]} and Z\(_{\odot }\) =20.8 pc [4]} for the position of the Sun and U\(_{\odot }\) =11.1, V\(_{\odot }\) =12.24, W\(_{\odot }\) =7.25  km.s\(^{-1}\)  and V\(_{LSR}\) =236.26  km.s\(^{-1}\)  ([5]} [5]} and [7]} [7]}).
